2-SYMMETRIC LOCALLYCONVEX SPACES
D. E. EDMUNDS In [l] it is shown that barrelledness and quasi-barrelledness are merely the two extreme examples of a property, called 2-symmetry, which may be possessed by a locally convex Hausdorff topological vector space. The object of this note is to show how recent char- acterisations [2; 3] of barrelled and quasi-barrelled spaces may be subsumed under characterisations of 2-symmetric spaces, and to ex- hibit some properties of these spaces. First we need some definitions and simple results. 1. Let £ be a locally convex Hausdorff topological vector space (abbreviated to LCS in what follows), and let S be a class of bounded subsets of E whose union is E. Let E' denote the topological dual of E, and let E'z be the set E' endowed with the topological of uniform convergence on the members of 2. Definition 1. A subset of E is said to be X-bornivorous if it absorbs every member of 2. Definition 2. We say that E is "2-symmetric if any of the following equivalent conditions hold : (a) Every S-bornivorous barrel in £ is a neighbourhood of zero. (b) Every bounded subset of E'j¡ is equicontinuous. (c) The topology induced on E by the strong dual of E% is the original topology of E. The equivalence of these conditions was proved in [l]. If 2iG22 it is easy to see that Si-symmetry implies 22-symmetry ; the strongest restriction on E is obtained by taking for 2 the class s of all subsets of E consisting of a single point, and then S-symmetry is simply the property of being barrelled. If 2 is the class b of all bounded subsets of E we have the weakest 2-symmetric property, which is that of being quasi-barrelled. Whatever the choice of 2, the topology of a 2-symmetric space is the Mackey topology [l]. Definition 3. E is said to be 2-bornological il every convex 2- bornivorous subset of E is a neighbourhood of zero. It follows easily that E is 2-bornological if and only if every linear map of E into an LCS F which takes members of 2 into bounded sets in F is continuous. For all choices of 2, a 2-bornological space is bornological. Given any LCS E and any class 2 of bounded sets whose union is Received by the editors August 13, 1962. 697
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E, we may define on £ a new topology 3 by taking as a fundamental system of neighbourhoods of zero the class of all convex circled 2- bornivorous subsets of E. 3 is the finest locally convex topology for which the members of S remain bounded, and the space F obtained by endowing the point set E with 3 is S-bornological. F is called the S-bornological space associated with E. We note that when S = s, 3 becomes the finest locally convex topology on E, and when S = &, 3 is the associated bornological topology [4, Chapitre 3, §2, Exercice 13]. Definition 4. Let Ei, E2 be LCS. A linear mapping u from £i onto E2 is said to be almost open if for every neighbourhood U of zero in Ei, the closure of m(í/) in E2 is a neighbourhood of zero for the Mackey topology on E2. Definition 5. A linear subspace Q oí the dual E' of an LCS E is said to be almost closed ii for every neighbourhood U of zero in E, U°r\Q is weakly closed in E', where U° denotes the polar of U. 2. The theorems in this section give characterisations of 2-sym- metric spaces. The first result includes Theorems 2.3 and 2.4 of [2] as special cases. Theorem 1. Let E be an LCS and 2 a class of bounded subsets of E whose union is E. Let F be the H-bornological space associated with E. Then E is ^-symmetric if and only if the topology of E is the Mackey topology and either of the following conditions holds : (a) The identity map from F onto E is almost open. (b) £' is almost closed in F'. Proof. This follows the pattern of the corresponding proof in [2 ] ; we give some details for convenience. To prove that 2-symmetry implies (b), all we need show, since F has the Mackey topology, is that E'C\K is weakly closed in F' for every weakly compact convex circled subset K of F'. Since K is an equicontinuous subset of F' it is bounded in F¿, so that ET\K is bounded in E'z, E'z being plainly a topological subspace of F¿. Since E is 2-symmetric, ET\K is thus an equicontinuous subset of £', and is hence relatively weakly com- pact in £'. Actually E'C\K is weakly compact in £', since E', is a topological subspace of F'„ and K is weakly closed in F'. It follows that KC\E' is weakly closed in F'. Condition (a) and the Mackey condition imply S-symmetry, since if U is a 2-bornivorous barrel in £ it is the closure in £ of a neighbour- hood of zero in F, so that by (a), U is a neighbourhood of zero in £. This completes the proof of'the theorem, since Pták [5] has shown that (a) and (b) are equivalent.
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The next theorem is a characterisation in terms of the closed graph theorem, and contains Theorems 2.2 and 3.1 of [3] as particular cases. Theorem 2. Let E, 2 be as in Theorem 1. Then E is 2-symmetric if and only if for every Banach space F the following is true: If u is any linear map from E into F such that (a) u takes members of 2 into bounded sets in F; (b) the graph of u is closed; then u is continuous. The proof is an obvious modification of that given in [3]. 3. We conclude by indicating various general properties of 2- symmetric and 2-bornological spaces. Theorem 3. (a) If E is 2-symmetric, E? is quasi-complete, (b) Let 2 be a class of convex, circled, closed bounded sets whose union is E. Then if E is 2-bornological, E's is complete. Proof, (a) Let B be a bounded closed subset of E'z. Since E is 2-symmetric, B is equicontinuous and is therefore complete [4, Chapitre 3, §3, Théorème 4]. (b) The completion of E's is the set of all linear functionals on E whose restriction to each member of 2 is continuous [4, Chapitre 4, §3, Exercice 3]. Such functionals are bounded on the members of 2, and since E is 2-bornological they are continuous on E. Hence E'z coincides with its completion. Specialisations of this theorem give, for example, the familiar re- sults that the dual of a quasi-barrelled space is strongly quasi-com- plete, and that the strong dual of a bornological space is complete. Theorem 4. Let (P¿)
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By choosing the 2< suitably we can obtain as special cases of this theorem the well-known results that the product of any family of barrelled (resp. quasi-barrelled) spaces is barrelled (resp. quasi-bar- relled.) Theorem 5. Let E be 2-symmetric and let M be a closed subspace of E. Denote by (p the canonical mapping E —»E/M, and put 2i = {(piS): SE?}. Then E/M is 'Zi-symmetric. The proof is obvious. References 1. S. Warner and A. Blair, On symmetry in convex topological vector spaces, Proc. Amer. Math. Soc. 6 (1955), 301-306. 2. M. Mahowald and G. Gould, Quasi-barrelled locally convex spaces, Proc. Amer. Math. Soc. 11 (1960), 811-816. 3. M. Mahowald, Barrelled spaces and the closed graph theorem, J. London Math. Soc. 36 (1961), 108-110. 4. N. Bourbaki, Espaces vectoriels topologiques, Chapitres 3-5, Hermann, Paris, 1955. 5. V. Pták, Completeness and the open mapping theorem, Bull. Soc. Math. France 86 (1958), 41-74. 6. A. Grothendieck, Espaces vectoriels topologiques, 2nd ed., Sociedade de Mate- mática de Säo Paulo, Säo Paulo, 1958. University College, Cardiff, Wales
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